↑

Properties

Label 1-4235-4235.3324-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.445 + 0.895i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.941 + 0.336i)2-s + (0.309 − 0.951i)3-s + (0.774 + 0.633i)4-s + (0.610 − 0.791i)6-s + (0.516 + 0.856i)8-s + (−0.809 − 0.587i)9-s + (0.841 − 0.540i)12-s + (0.998 + 0.0570i)13-s + (0.198 + 0.980i)16-s + (0.466 + 0.884i)17-s + (−0.564 − 0.825i)18-s + (−0.985 − 0.170i)19-s + (−0.415 + 0.909i)23-s + (0.974 − 0.226i)24-s + (0.921 + 0.389i)26-s + (−0.809 + 0.587i)27-s + ⋯
L(s)  = 1  + (0.941 + 0.336i)2-s + (0.309 − 0.951i)3-s + (0.774 + 0.633i)4-s + (0.610 − 0.791i)6-s + (0.516 + 0.856i)8-s + (−0.809 − 0.587i)9-s + (0.841 − 0.540i)12-s + (0.998 + 0.0570i)13-s + (0.198 + 0.980i)16-s + (0.466 + 0.884i)17-s + (−0.564 − 0.825i)18-s + (−0.985 − 0.170i)19-s + (−0.415 + 0.909i)23-s + (0.974 − 0.226i)24-s + (0.921 + 0.389i)26-s + (−0.809 + 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.445 + 0.895i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (3324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.445 + 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.799491468 + 1.733076008i\)
\(L(\frac12)\) \(\approx\) \(2.799491468 + 1.733076008i\)
\(L(1)\) \(\approx\) \(1.959318894 + 0.2474309331i\)
\(L(1)\) \(\approx\) \(1.959318894 + 0.2474309331i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.941 + 0.336i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
13 \( 1 + (0.998 + 0.0570i)T \)
17 \( 1 + (0.466 + 0.884i)T \)
19 \( 1 + (-0.985 - 0.170i)T \)
23 \( 1 + (-0.415 + 0.909i)T \)
29 \( 1 + (-0.897 + 0.441i)T \)
31 \( 1 + (0.362 + 0.931i)T \)
37 \( 1 + (0.254 - 0.967i)T \)
41 \( 1 + (-0.0285 + 0.999i)T \)
43 \( 1 + (-0.654 + 0.755i)T \)
47 \( 1 + (-0.564 + 0.825i)T \)
53 \( 1 + (-0.198 + 0.980i)T \)
59 \( 1 + (0.0285 + 0.999i)T \)
61 \( 1 + (0.941 - 0.336i)T \)
67 \( 1 + (0.959 - 0.281i)T \)
71 \( 1 + (0.696 + 0.717i)T \)
73 \( 1 + (-0.993 - 0.113i)T \)
79 \( 1 + (-0.0855 - 0.996i)T \)
83 \( 1 + (0.870 + 0.491i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + (0.974 - 0.226i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.725260835319741907994121782446, −17.355662303368459025181521285254, −16.55945255296969709895086112722, −16.1307965337577247249713547621, −15.358710415092444341220966176024, −14.88217489557549730243481405893, −14.20728497243499750727772683226, −13.54116099614983717519328917257, −13.01442260996697778758200074921, −11.997939907451966086931905819980, −11.41360462846528355336432984326, −10.792282858180619107164463763863, −10.09459680125951225279562307029, −9.57052606331061603094531188134, −8.55914159795041471761067106973, −7.969528659730733754088323642912, −6.829754830175432647708973736579, −6.131976892494543023057268543413, −5.37892759087248103762407636259, −4.74634346649375495614692142309, −3.86581739390511766142375067368, −3.56027576146736246100515123213, −2.516548961097632758071936326148, −1.94858997524124133106215582610, −0.561915156882174119219162645899, 1.32046625947396854593893050513, 1.83477049784948736302365077347, 2.84594084817579903529705614026, 3.53623456510146598517633585605, 4.1817041314204695048409699952, 5.284331958273592754881866039443, 6.04573229367177910605622182306, 6.42434946521543083729260040218, 7.26384626705877800610694578211, 7.99546810109491249452064249402, 8.45901797257981836649963774545, 9.320520850475430070989486099553, 10.55947181620593499993882530933, 11.20259554489982429461414423003, 11.84737210295705611498296929928, 12.68468137650755080248870356907, 13.04418763291724659618289484328, 13.64002443054615401363125338251, 14.49081583400010562266087864234, 14.79062878200231020441948417518, 15.69732870208926487321720343030, 16.35814853936390087887874178076, 17.17559674236598721646530302125, 17.70250489776499327032068054683, 18.49143935184752277241929299870

Graph of the $Z$-function along the critical line